Is it a "criminal record check with a vulnerable sector search" or a background check? There's a difference.
The former is usually for jobs with vulnerable populations (e.g. healthcare, childcare, seniors, etc). All you need is provide proof of address, photo ID, have your fingerprints taken at the police station.
The later often includes employment and reference checks, etc and may be more detailed. This is the type people undergo for federal/provincial gov't jobs, high level security clearance jobs, etc.
Is there a reason they can not begin the checks now while waiting for your PR paperwork? The cost is usually paid by the would-be employee anyway.
Is this what the employer is asking: http://www.rcmp.gc.ca/cr-cj/steps-etapes-eng.htm
Processing times: http://www.rcmp.gc.ca/cr-cj/time-temps-eng.htm